↑

Properties

Label 1-2888-2888.2851-r1-0-0
Degree $1$
Conductor $2888$
Sign $0.487 + 0.873i$
Analytic cond. $310.358$
Root an. cond. $310.358$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.245 − 0.969i)3-s + (0.986 + 0.164i)5-s + (0.0825 + 0.996i)7-s + (−0.879 − 0.475i)9-s + (−0.879 + 0.475i)11-s + (−0.245 + 0.969i)13-s + (0.401 − 0.915i)15-s + (−0.0825 − 0.996i)17-s + (0.986 + 0.164i)21-s + (−0.245 − 0.969i)23-s + (0.945 + 0.324i)25-s + (−0.677 + 0.735i)27-s + (0.0825 + 0.996i)29-s + (0.677 − 0.735i)31-s + (0.245 + 0.969i)33-s + ⋯
L(s)  = 1  + (0.245 − 0.969i)3-s + (0.986 + 0.164i)5-s + (0.0825 + 0.996i)7-s + (−0.879 − 0.475i)9-s + (−0.879 + 0.475i)11-s + (−0.245 + 0.969i)13-s + (0.401 − 0.915i)15-s + (−0.0825 − 0.996i)17-s + (0.986 + 0.164i)21-s + (−0.245 − 0.969i)23-s + (0.945 + 0.324i)25-s + (−0.677 + 0.735i)27-s + (0.0825 + 0.996i)29-s + (0.677 − 0.735i)31-s + (0.245 + 0.969i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.487 + 0.873i$
Analytic conductor: \(310.358\)
Root analytic conductor: \(310.358\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2888,\ (1:\ ),\ 0.487 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774259011 + 1.041597400i\)
\(L(\frac12)\) \(\approx\) \(1.774259011 + 1.041597400i\)
\(L(1)\) \(\approx\) \(1.211792842 - 0.09227850427i\)
\(L(1)\) \(\approx\) \(1.211792842 - 0.09227850427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.245 - 0.969i)T \)
5 \( 1 + (0.986 + 0.164i)T \)
7 \( 1 + (0.0825 + 0.996i)T \)
11 \( 1 + (-0.879 + 0.475i)T \)
13 \( 1 + (-0.245 + 0.969i)T \)
17 \( 1 + (-0.0825 - 0.996i)T \)
23 \( 1 + (-0.245 - 0.969i)T \)
29 \( 1 + (0.0825 + 0.996i)T \)
31 \( 1 + (0.677 - 0.735i)T \)
37 \( 1 + (0.879 - 0.475i)T \)
41 \( 1 + (-0.401 + 0.915i)T \)
43 \( 1 + (0.546 - 0.837i)T \)
47 \( 1 + (0.879 - 0.475i)T \)
53 \( 1 + (0.879 - 0.475i)T \)
59 \( 1 + (-0.401 + 0.915i)T \)
61 \( 1 + (-0.789 - 0.614i)T \)
67 \( 1 + (0.789 - 0.614i)T \)
71 \( 1 + (-0.789 + 0.614i)T \)
73 \( 1 + (-0.0825 - 0.996i)T \)
79 \( 1 + (-0.546 + 0.837i)T \)
83 \( 1 + (-0.986 + 0.164i)T \)
89 \( 1 + (-0.0825 + 0.996i)T \)
97 \( 1 + (0.789 + 0.614i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.951135261556590447314105429062, −17.87508586437645068287446331773, −17.25935435355167836177211723385, −16.95723619617117298446857996261, −15.93677789866248971966844041611, −15.46703318330251320751804182952, −14.583550563853250808244585924756, −13.890542150991364201030390971330, −13.36388173185494893471562403443, −12.74156331547967165272443377849, −11.51005993866215680985639915846, −10.63877585084446457447267665501, −10.282717093904259368776000574770, −9.78959037322082670198618430983, −8.81614834216597294620351811660, −8.0972601913106213076340095429, −7.453966603884667048047893426014, −6.1177474831485243539587220175, −5.6730681211707319645119970719, −4.83179199372180542902865927799, −4.11010184991322817527271672902, −3.1627781122417282429551453214, −2.52764255681783575078670862372, −1.3745710935714739466027005749, −0.333095283999881463860188596247, 0.87958768458824084499756402852, 2.04238605286948911327341170222, 2.358453309388651181136286206047, 3.009072688393850528130059993040, 4.53161024495977634880495333288, 5.32564672172773551929051758640, 6.00945126191157496192733276322, 6.74660083306671439277608951472, 7.37519093328011537262925316055, 8.32351784048387320141213322774, 9.0469087654234168346575520898, 9.56851209395842130456785065092, 10.499719163134933635225799223301, 11.457350002419399813193140248966, 12.13126569774579532595797669256, 12.74979794788521916538487600081, 13.43822325173646776070146419582, 14.11187410365269277762384645983, 14.67474885005454121925205492194, 15.4449399415149774058113497771, 16.43364840847130876203859038033, 17.100794562626244030120412432926, 18.06522593172281752044905289723, 18.37178038222674912334395971854, 18.716439340964867052360232555301

Graph of the $Z$-function along the critical line